Load Flow in electrical power systems - Stuttgart Regionsix6.region-stuttgart.de/sixcms/media.php/773/LOADFLOW_Tartaglia.pdf · Load Flow in electrical power systems Michele ... Where

Warsaw Summer School SUSTAINABLE URBAN ENERGY CONCEPTS 31 August - 4 September 2009

Load Flow Load Flow in electrical power systemsin electrical power systems

Michele Tartaglia, Lazzeroni PaoloPolitecnico di Torino - Dipartimento di Ingegneria Elettrica

Corso Duca degli Abruzzi 24 - 10129 Torino, Italy

Tel: +39-011-5647110; fax +39-011-5647199


Load Flow analisysLoad Flow analisysAn electric network with An electric network with nn buses can be represented by matrix buses can be represented by matrix equation:equation:

Where Where ii and and vv are the current and voltage vectors respectively. are the current and voltage vectors respectively. YYbusbus represents the admittance matrix in which represents the admittance matrix in which YY1111--YYnn nn represents represents the sum of admittances connected between node and ground, the sum of admittances connected between node and ground, while the other elements represents elements connecting two while the other elements represents elements connecting two different nodesdifferent nodes

vYi bus =

nnnn

n

n V

V

YY

YY

I

I

M

L

MOM

L

M

1

1

1111

=


Load Flow analisysLoad Flow analisysThe current in a generic node The current in a generic node ii can be expressed by:can be expressed by:

The complex power at the same node is equal to:The complex power at the same node is equal to:

Where Where PPGGii and and CCii are the produced and adsorbed active power are the produced and adsorbed active power at node at node ii respectively. respectively. QQGiGi and and DDii are the produced and adsorbed are the produced and adsorbed reactive power at node reactive power at node ii..

ik

n

kki YVI =

=1

( )iGiGn

kikiiiii DQjCPYVVIVS ii +===

=1

***


Load Flow analisysLoad Flow analisysThe equation of complex power can be also expressed using The equation of complex power can be also expressed using polar coordinates:polar coordinates:

This equation can be dived in real and imaginary part:This equation can be dived in real and imaginary part:

This nonThis non--linear equation can be solved using Newtonlinear equation can be solved using Newton--Raphson Raphson method to find the module and phase of voltage for each node.method to find the module and phase of voltage for each node.

( ) ( )=

=+n

k

jikkiiGiG

ikki

iieYVVDQjCP

1

( )

( ) 0sin

0cos

1

1

=

=

=

=

n

kikkiikkiiG

n

kikkiikkiiG

YVVDQ

YVVCP

i

i


Load Flow analisysLoad Flow analisysBefore to apply the NewtonBefore to apply the Newton--Raphson method the nodes are Raphson method the nodes are divided un three different categories:divided un three different categories:

PQ nodesPQ nodes: where are connected pure loads (: where are connected pure loads (PPGiGi=0 , =0 , QQGiGi=0)=0)

PV nodesPV nodes: where are connected generators (the value of : where are connected generators (the value of PPGiGiand and VVii are known)are known)

Slack nodeSlack node: balance node of the system (module and phase of : balance node of the system (module and phase of the voltage are known)the voltage are known)


Load Flow analisysLoad Flow analisysStarting from these categories of nodes is possible to define thStarting from these categories of nodes is possible to define the e number of equation to calculate module and phase of voltage for number of equation to calculate module and phase of voltage for each node.each node.

PQ nodesPQ nodes: introduces 2 equation (the real and imaginary : introduces 2 equation (the real and imaginary equation of the complex power) because in that nodes module equation of the complex power) because in that nodes module and phase of the voltage are unknown.and phase of the voltage are unknown.

PV nodesPV nodes: introduces 1 equation (the real equation of the : introduces 1 equation (the real equation of the complex power) because in that nodes module is known complex power) because in that nodes module is known (imposed by the generator) and phase of the voltage is (imposed by the generator) and phase of the voltage is unknownunknown

Slack nodeSlack node: doesn: doesnt introduce equationt introduce equation


Load Flow analisysLoad Flow analisysThe NewtonThe Newton--Raphson method is a numerical approach that can Raphson method is a numerical approach that can be used to determine the root of a function f(x) (monodimentionabe used to determine the root of a function f(x) (monodimentional l for example).for example).

In particular if is known an approximated value of the solution In particular if is known an approximated value of the solution xx00

is possible to represent the same equation f(x) by its first oris possible to represent the same equation f(x) by its first order der Taylor series, around the point x0:Taylor series, around the point x0:

( ) ( ) ( )00'0)( xxxfxfxf +=


Load Flow analisysLoad Flow analisysThis equation represents also the tangent straigth line at the This equation represents also the tangent straigth line at the function f(x) in xfunction f(x) in x00. If we extended the line as far as to intersect . If we extended the line as far as to intersect the x axes we find a new approximate solution xthe x axes we find a new approximate solution x11

Starting from this point is possible to realize an iterative Starting from this point is possible to realize an iterative procedure to calculate the solution with a stop criteria:procedure to calculate the solution with a stop criteria:

x1

f(x)

x0x2

x*

x3

( )


Load Flow analisysLoad Flow analisysIf the function f(x) is multidimensional the first order Taylor If the function f(x) is multidimensional the first order Taylor series becams equal to:series becams equal to:

Where J is the Jacobian matrix:Where J is the Jacobian matrix:

The iterative procedure can be also modified:The iterative procedure can be also modified:

)()()( 111 += hhhxhh xxJxfxf

[ ] )( 1111 = hhxhh xfJxx

n

nn

n

x

x

f

x

f

x

f

x

f

J

=

LL

MOM

MOM

LL

1

1

1

1

{ }


Load Flow analisysLoad Flow analisysExample: 2 nodes with one generatorExample: 2 nodes with one generator

GeneratorLineNetwork

jX

PG,QG

01

10

jX

jXYbus

=

Node 1 Node 2

( )

02

cos

0cos

221

1

=

+

= =

VVP

YVVCP

G

n

kikkiikkiiGi


DIgSILENT applicationDIgSILENT applicationMichele Tartaglia, Lazzeroni Paolo

Politecnico di Torino - Dipartimento di Ingegneria Elettrica

Corso Duca degli Abruzzi 24 - 10129 Torino, Italy

Tel: +39-011-5647110; fax +39-011-5647199


DIgSILENTDIgSILENTThe following exercitation is related to the MV electrical netwoThe following exercitation is related to the MV electrical network rk that supply the Arquata district in Turin.that supply the Arquata district in Turin.

It will be analyzed the impact of a cogenerator with different It will be analyzed the impact of a cogenerator with different electrical power level produced, in the MV network:electrical power level produced, in the MV network:

1 MW (the real situation)1 MW (the real situation)

10 MW10 MW

100 MW100 MW


DIgSILENTDIgSILENTThe DIgSILENT software allows to build all the components The DIgSILENT software allows to build all the components which are present in the networkwhich are present in the network


DIgSILENTDIgSILENTCase 1: 1 MWCase 1: 1 MW

Voltage profile node 203565

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Load Flow in electrical power systems - Stuttgart Regionsix6.region-stuttgart.de/sixcms/media.php/773/LOADFLOW_Tartaglia.pdf · Load Flow in electrical power systems Michele ... Where